The competition between thermal fluctuations and potential forces governs the stability of matter in equilibrium, in particular the proliferation and annihilation of topological defects. However, driving matter out of equilibrium allows for a new class of forces that are neither attractive nor repulsive, but rather transverse. The possibility of activating transverse forces raises the question of how they affect basic principles of material self-organization and control. Here we show that transverse forces organize colloidal spinners into odd elastic crystals crisscrossed by motile dislocations. These motile topological defects organize into a polycrystal made of grains with tunable length scale and rotation rate. The self-kneading dynamics drive super-diffusive mass transport, which can be controlled over orders of magnitude by varying the spinning rate. Simulations of both a minimal model and fully resolved hydrodynamics establish the generic nature of this crystal whorl state. Using a continuum theory, we show that both odd and Hall stresses can destabilize odd elastic crystals, giving rise to a generic state of crystalline active matter. Adding rotations to a material's constituents has far-reaching consequences for continuous control of structures and transport at all scales.

The addition of transverse forces to an ensemble of colloidal spinners induces the appearance of odd elastic crystals, featuring self-propelled defects that organize the system into a 'self-kneading' crystal whorl state.